Fun math algorithms

[Albert Chan]

(10-17-2020 08:51 AM)EdS2 Wrote:  I don't understand where the cubic comes from - please elaborate, if you can!

My guess is you were really asking where C comes from.

You can think of car payments like this:
Instead of paying the loan, put car payments into a bank, collecting interest (same rate).
At the end of terms, take out the money, and pay off the loan, all at once.

At the end of terms, your bank deposit have grown (last deposit have no interest):

-FV = PMT*(1+I)^(N-1) + PMT*(1+I)^(N-2) + ... + PMT

  |FV/PMT| = 1 + (1+I) + ... + (1+I)^(N-1)
(1+I)|FV/PMT| = (1+I) + ... + (1+I)^(N-1) + (1+I)^N

Subtract 2 expressions, we have: I |FV/PMT| = (1+I)^N - 1

Since we put off paying the loan, it has grown too, FV = PV*(1+I)^N
Both FV must match, we can simply substitute:

I |PV*(1+I)^N/PMT| = (1+I)^N - 1
I |PV/PMT| = 1 - (1+I)^-N

C = |N*PMT/PV| = I*N / (1 - (1+I)^-N)

Nice thing about C is, for small N, it is relatively flat, C-1 ≈ I*(N+1)/2
(if I=0, we have C-1=0, or no financing cost, as expeced)

Once we calculated C, we can ignore interest rate: N |PMT| = C |PV|